The 25-year-old looked to have secured the green jacket with a birdie at the 15th, which took him to nine-under par with four holes left to play.
However, a bogey at 16 and Adam Scott's birdie at 15 saw him drop back into a share, before another dropped shot at 17 took him out of contention.
Despite disappointment, Day said that he has taken a lot of experience from his week at Augusta and that he will be back to challenge next year.
"It's really tough," Day told reporters. "The pressure got to me a little bit.
"It's a little disappointing, but there's a lot of experience that I can take into the next year, and hopefully I can wear one of those green jackets soon.
"I love this Tournament regardless of where I finish, it's an honor to come and play against the best players in the world."